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Questions tagged [complex-integration]

This is for questions about integration methods that use results from complex analysis and their applications. The theory of complex integration is elegant, powerful, and a useful tool for physicists and engineers. It also connects widely with other branches of mathematics.

The concept of definite integral of real functions does not directly extend to the case of complex functions, since real functions are usually integrated over intervals but complex functions are integrated over curves.

Surprisingly complex integration are not so complex to evaluate, oftenly simpler than the evaluation of real integrations. Some real integrations which are otherwise difficult to evaluate can be evaluated easily by complex integration, and moreover, some basic properties of analytic functions are established by complex integration only.

References:

https://en.wikipedia.org/wiki/Contour_integration

"An Introduction to the Theory of Analytic. Functions of One Complex Variable" by Lars Ahlfors

"Complex Variables and Applications " by James Ward Brown and Ruel V. Churchill

3109 questions
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how to solve it by residue theorem

i have to solve the folowing integral by using contour integration & residue theorem $$\int_{-\infty}^{\infty}\frac{\sin x}{x^2+2x+2}\ dx$$ i factorised $x^2+2x+2=(x+1+i)(x+1-i)$ where $x=-1+i$ & $x=-1-i$ are single poles…
Bhaskara-III
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Complex integration with $\epsilon$ and $\pi$.

I'm not sure how to solve this integral: $$ \int_\gamma \frac{dz}{(e^z+4)(z-\pi i)}, $$ where $ \gamma$ is the region $ \|z-\sqrt7i\|+\|z+\sqrt7i\|=8$. I know that region is the ellipse $\frac{x^2}{9}+\frac{y^2}{16}=1 $ but I don't know how to…
Renato Collado Tello
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Evaluation of complex integral

$$\int_{\text{c}}\frac{\sin \pi z^2 + \cos \pi z^2}{(z+1)(z+2)}$$ Where $\text{C}$ is the circle $|z| =3$ I'm a little confused about how to do this. Should this be done the normal way ? How do I use the concept of what C is and the concepts of…
Saikat
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Matusubara sum contradiction?

In many textbooks, the following fermionic Matsubara sum is given as a useful identity: $$T\sum_{\omega_n}\frac{1}{i\omega_n-\epsilon}=\frac{1}{e^{\epsilon/T}+1},$$ where $\omega_n=n\pi T$ for all odd $n$. However, when $\epsilon=0$, the LHS…
anon
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Integration with Jordan's Lemma

I'm trying to integrate the following: $$I = \int^\infty_0 \dfrac{\cos x}{x^2 + 1} dx$$ What I did was: $$I = \int^\infty_0 \dfrac{\cos x}{x^2 + 1} dx = \dfrac{1}{2}\int^\infty_{-\infty} \dfrac{e^{ix}+e^{-ix}}{2(x^2+1)} dx =…
iamatrain
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Is $\int_{0}^\infty \frac{\sin(nx)}x \,dx$ is equal to $\pi/2$?

Is $\int_{0}^\infty \frac{\sin(nx)}x \,dx$ is equal to $\pi/2$ for positive real $n$? I've come to this answer by inverse Fourier transform. But since there is n, I am quite confused that I didn't get n in the answer. Is this answer…
Nikhanbayev Nursultan
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Complex integral difficulty

We need some sort of analytic expression for the integral: $$\int^\infty _{-\infty} \frac{\sqrt{\alpha^2 + 1}}{(\mathrm{i}\alpha)^{\frac{3}{4}}}\mathrm{e}^{\mathrm{i}\alpha \chi} \mathrm{d}\alpha$$ where $\chi$ is a real number. Any thoughts? EDIT…
QMRush
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Complex integral using Residue Theorem with a regularised pole

I need to prove the following integral computation by applying the residue theorem: $$\int_{-\infty}^{+\infty}ds \ e^{-i\Omega s}\bigg(\frac{a^2}{4(\sinh{[\frac{a}{2}s]})^2}-\frac{1}{s^2}\bigg)=-2\pi\Omega\frac{1}{e^{\frac{2\pi\Omega}{a}}-1}$$ where…
Fischer
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Complex integral computation with $\sinh$

I need to prove the following integral computation by applying the residue theorem: $$\int_{-\infty}^{+\infty}ds\frac{e^{-i\Omega s}}{(\sinh{[\frac{a}{2}s-i\epsilon]})^2}=-8\pi\frac{\Omega}{a^2}\frac{1}{e^{\frac{2\pi\Omega}{a}}-1}$$ As far as I…
Fischer
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Evaluate the integral along a contour containing 2 interior points by using Cauchy's Integral Formula

Evaluate the integral $$\int_{C}\frac{z^2}{z^2+9}dz$$ where C is the circle $|z|=4$ I know that if f is analytic in simply connected domain $D$, $C$ a simple closed positively oriented contour that lies in D and $z_o$ lies interior to $C$,…
Wang Kah Lun
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What is the meaning of this integral?

Does anyone know the meaning of this type of integral? $\displaystyle{\int f(z) \,\overline {dz}}$. I think this means $\displaystyle{\int u\,dx + v\,dy+i\int v\,dx -u\,dy}$ where $f=u + iv$
user194502
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one complex variables (integration)

how to prove $\int_{C_R}\frac{\log^3(z)}{(1+z^2)^2}\,dz$ goes to $0$ as $R$ goes to $\infty$, with $C_R=Re^{it}$ for $00$
Abdallah
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understanding a particular step in proof of cauchy's theorem for triangles

"Hi: I am reading "complex variables" by Ash and Novinger and they prove "cauchy's theorem for triangles early in the book". Unfortunately, there's a step in their proof that I don't follow. Assuming that $f(z)$ is analytic at $z_{0}$, they first…
mark leeds
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Complex integration problem

Im having a bit trouble getting started in complex integrals. Im integrating along a circle the integral $\int \frac{e^z}{z^2 + 1}$ where a) center O radius 1/2 b) center i radius 1 c) center i radius 3 I have set $\gamma(t) = \frac{1}{2}e^{it}$ and…
dingari
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Finding $ \int_{\gamma(0;1)} \frac{\text{Im}(z)}{z-\frac{1}{2}} \,dz $

I have a question that I cannot wrap my head around. How would I compute this integral using complex analysis techniques? $$ \int_{\gamma(0;1)} \frac{\text{Im}(z)}{z-\frac{1}{2}} \,dz $$ I was thinking about whether I need to consider another…
NeedHelpNotFailingMath
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